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THE EMBEDDED CONTACT HOMOLOGY OF S1 £ D2
by
Roman Golovko
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Ful¯llment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MATHEMATICS)
August 2009
Copyright 2009 Roman Golovko

The goal of this thesis is to understand generalizations of (cylindrical) contact homology and embedded contact homology to contact 3-manifolds with sutured boundary. Both contact homology and embedded contact homology count holomorphic curves in the symplectization of a contact 3-manifold, whose foundations were established by Hofer and Hofer-Wysocki-Zehnder.; Cylindrical contact homology is an invariant of a contact 3-manifold, and counts rational curves with one positive puncture and one negative punctures in the symplectization of a contact manifold. It is due to Eliashberg-Givental-Hofer as a special case of Symplectic Field Theory.; Embedded contact homology, introduced by Hutchings, roughly speaking, counts embedded holomorphic curves in the symplectization of a contact 3-manifold, which are asymptotic to periodic Reeb orbits. Embedded contact homology is a topological invariant and is conjecturally equivalent to Heegaard Floer homology.; In this thesis, we construct the first series of nontrivial examples where the sutured version of embedded contact homology coincides with sutured Floer homology. This result provides some evidence that the conjecture about the equivalence of the sutured version of embedded contact homology and sutured Floer homology holds. In addition, for the constructed sutured contact manifolds we calculate the sutured version of cylindrical contact homology.

THE EMBEDDED CONTACT HOMOLOGY OF S1 £ D2
by
Roman Golovko
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Ful¯llment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MATHEMATICS)
August 2009
Copyright 2009 Roman Golovko